A maximum entropy principle in general relativity and the stability of fluid spheres

A NNALES DE L ’I. H. P., SECTION A

W. J. C OCKE

A maximum entropy principle in general relativity and the stability of fluid spheres

Annales de l’I. H. P., section A, tome 2, n

^o

4 (1965), p. 283-306

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283

A maximum entropy principle

in general relativity

and the stability ^{of fluid} spheres (1)

W.

J. COCKE

(Cornell University (N. Y)

^{et Institut Henri}

Poincaré, Paris)

Ann. Inst. Henri Poincaré, Vol. II, ^no 4, 1965,

Section A :

Physique théorique.

SUMMARY. - The

principle

^{that the total}

entropy

^{be a} maximum is

applied

to

gravitational systems

^{of a}

perfect,

relativistic fluid in adiabatic motion.

An

expression

^{for the}

entropy density

^of the fluid is derived as an

explicit

function of the _energy

density,

^and the conservation

equations imply

^that

this

expression

^satisfies the proper

continuity equation.

^The

entropy principle

^{is shown to}

yield

^the

equation

^of

hydrostatic equilibrium

^{and a}

stability

^criterion

equivalent

^{to one derived from}

dynamics. Application

to features of

equilibrium

models is studied.

SOMMAIRE. - Le

principe d’entropie

^{maximum est}

applique

^au

système gravitationnel :

^fluide relativiste

parfait

^{en mouvement}

adiabatique.

On trouve une

expression

de la densite

d’entropie,

^fonction

explicite

^{de la}

densite

d’energie.

^Les

equations

de conservation entrainent _que cette fonc- tion satisfait

identiquement

^{a une}

equation

de continuité. On deduit du

principe d’entropie

^maximum

1’equation d’équilibre hydrostatique,

^{et un}

critère de stabilité

equivalent

^{a celui}

qui

découle de la

dynamique.

^Des

applications

^{aux modèles}

d’équilibre

^{sont etudiees.}

(1)

^This paper is a revised version of a thesis presented ^{to the}

Faculty

^{of the}

Graduate School of Cornell

University,

Ithaca, ^New York, ^{U. S.} A., ⁱⁿ

partial

fulfillment of the

requirements

^{for the} degree ^{of Doctor of}

Philosophy.

The author is

presently

the holder of a Postdoctoral

Fellowship

of the National Science Foundation.

ANN. INST. POINCARÉ, A-II-4 ¹⁹

284 W. J. COCKE

INTRODUCTION

This paper deals

chiefly

^with

setting

up ^a maximum entropy

principle

for

gravitationally

^bound

spheres

^of

perfect fluid,

^{treated via Einstein’s}

general theory

^of

relativity.

We will derive an

expression

for the total

entropy

^{of such a fluid}

sphere

^{in terms} of the pressure, energy

density,

and

gravitational potentials

^and

require

^{that this}

expression

^{be a maximum}

with

respect

^{to small}

perturbations conserving

^{the total energy of the fluid.}

We will show that this

requirement implies

both the relativistic

equation

of

hydrostatic equilibrium

^{and a} criterion for the

stability

^{of static}

configu-

rations of fluid

satisfying

^this

equation.

Considerable interest will lie in the

stability criterion,

and it will be shown that it is

equivalent

^{to one}

previously

^obtained

by

^S. Chandrasekhar from

dynamical

considerations. We will also use it to elucidate certain features of actual

equilibrium

^{models. In}

particular,

^we will consider how unstable

configurations

^{are related to} the location of maxima and minima of

plots

^of

equilibrium

^{mass curves,}

parametrized

^as functions of the central pressure ^or

density.

Throughout

^the paper ^we consider all motions to be

adiabatic,

^{so that}

we

neglect

^heat

flow,

^{but we make no other}

assumption

^{as to} the form of the

equation

^{of state,}

except

that it be a function of the usual

thermodynamic

variables. We will limit our discussion to

stability against spherically symmetric perturbations, although

the formalism could be extended to include other

types.

In

general

^{there are two} methods of

testing

^the

stability

^{of a} static confi-

guration

^of

fluid,

both of which have been

applied

^{to the Newtonian case.}

We will ^now

briefly

discuss these two methods and indicate the connection between them.

The first method is known as the «

dynamical

method » and has been

applied

^to both Newtonian and relativistic fluid mechanics

[1, 2].

^It

involves

assuming

^{that the}

quantities 03C8j characterizing

^{the state} of the fluid

system deviate,

^as functions of

positions

^and

time, only

very

slightly

^from

.a set of

given

^static

quantities

^One may then take the

time-dependent equations

of motion to be linear

equations

with the small differences

~ 2014

⁼⁼

~

^as

^dependent

^variables.

If the motions are assumed to be

adiabatic,

^{and hence}

reversible,

^the

time-dependance

^{can be}

separated

^out

by assuming

^a functional

dependence

on r, t of the form

~y

^{= cos} at, where ^a is a constant. The

equations

285

A MAXIMUM ENTROPY PRINCIPLE

for

~~~(r)

^can then be transformed into ^a Sturm-Liouville

eigenvalue

pro- blem, ^{with a2 as}

eigenvalue.

^{See in}

particular

section I of this _paper, where

we

present

^{a sketch of the}

dynamical

^{method in}

^general relativity.

In

general

^the admissible

eigenvalues

^{a2 form a}

discrete, countably

^infinite

set, ^and

stability

is tested

by ascertaining

^{whether or not} all the a2’s are

positive.

^{If not, then some} of the a’s are

imaginary,

^and excitation of these modes will result in an

exponential time-dependence

instead of ^an oscil-

latory

^{one. This} will carry the

configuration

^{so far} away from the

given

static state that the assumed linear

approximation

will become invalid.

A static

configuration

is then stable with

respect

^{to small}

perturbations only

if all the

eigenvalues s2

associated with it ^are

positive.

The second method is known as the ^« _energy method » and has so far

only

^been

applied

in Newtonian

theory [3, 4, 5].

^{It states the}

following:

if the total conserved energy E of ^{a mass} of fluid can be written as a sum

E ⁼ T +

U,

^{where T is a}

positive

definite kinetic _energy and U is a

poten-

tial _energy which does not

depend

^{on the}

hydrodynamic velocities,

^{then a}

configuration

is in stable

equilibrium only

^{if U is a minimum with}

respect

to infinitesimal variations of the

quantities ~j

^{which conserve the total}

amount

of

^material

(in

^Newtonian

theory,

the total

mass).

^{That this is}

necessary for the

stability

^{of a} static distribution of material is

evident,

since otherwise _any

lessening

^{of the}

potential

energy U

by

^{a small}

pertur-

bation would release kinetic _energy and

provide

^further

impetus

^{to the}

displacement.

In terms of the calculus of

variations, requiring

^{U to be a minimum means}

that ^we must have 3D ⁼ 0 and ö2U &#x3E;

0,

^{where ~U is the} first variation and ö2U is the second variation. au ⁼ 0

implies

^{an Euler}

equation,

^which

turns out to be

simply

^the

hydrostatic equilibrium equation,

^{and a2D} &#x3E; 0 is then a

stability

^criterion.

The connection between the

dynamical

^{method and the} energy method is made as follows : it turns out

[6]

^{that a} necessary and sufficient condition for 82U &#x3E; 0 for non-zero variations is that the

eigenvalues

fL of a

particular eigenvalue problem

^{all be}

positive.

^This

eigenvalue problem

^{is in}

general

not

quite

^{the same as the one} associated with the

dynamical

^method

(as

discussed

above),

^{but one} should be able to show that all

the , ’s

^are

positive

if and

only

if all the a2’s are

positive.

This would show that the two stabi-

lity

^{criteria are} in fact the same. This has been done in the Newtonian

case

[5, 6].

In

general relativity, however,

the formulation of the _energy method is

complicated by

^the

apparent

^{lack of a}

quantity characterizing

^{the « total}

286 W. J. COCKE

amount of material ». In Newtonian mechanics this

quantity

^is

simply

the total ^mass of the

fluid,

^{but in}

relativity

the fundamental

equivalence

of mass and _energy

implies

^{that one} could alter the total ^mass of a

body

without

actually removing

^or

adding

any ^« material».

Many

^authors

have used the

concept

^{of a total}

particle

^{number or so-called «} proper mass », but this _usage ^seems

foreign

^to

hydrodynamics,

^{which is a}

theory

^{of conti-}

nuous fields

[7].

We _may,

however,

^{turn to the}

thermodynamic aspect

^{of the}

problem

and

investigate

^the

requirement

that the total

entropy

^{of the}

system

^{be a}

maximum for

stability.

It is this

principle which,

^{as stated}

above,

^{we will}

develope

^{in this} paper, and with which we propose ^to

replace

^{the minimum}

potential

^{energy method.}

Since we consider

only

^adiabatic

motions,

^our

expression

for the total

entropy

^{S will} turn out to be ^a constant of the motion. We will derive

an

expression

^{for the}

entropy density

^{in terms of}

quantities entering directly

into Einstein’s field

equations,

^{and then}

verify

from the- field

equations

that it satisfies the _proper

continuity equation.

^An

expression

^{for the conser-}

ved total

entropy

^{S will} then follow. This will

give

^{us a} non-trivial

integral

of Einstein’s field

equations

^for

general

adiabatic fluid motions.

We will next show that the statement 8S ⁼

0,

taken relative to variations which conserve the total _energy,

implies

the relativistic

equation

^of

hydro-

static

equilibrium.

The additional maximum

requirement

^{82S 0 will}

then be shown to lead to the same

stability

criterion obtained

by

^Chan-

drasekhar from the

dynamical

method. In the last

section,

^{the maximum} entropy

principle

^{will then be}

applied

^{to some other}

interesting

ques- tions.

At this

point

^we

might

say ^a few words

regarding

^the

assumption

^that

no heat flow occurs. It is of course evident that this

postulate

^{is never}

exactly

satisfied in nature, but ^we may

regard

^{it as}

approximately

^{true over}

appropriately

short intervals of proper time relative ^{to an} observer

moving

with the fluid. In situations

involving

very strong

gravitational fields,

such as in a

gravitational collapse problem,

^this

approximation

^would

appear ^to be a _propos, in as much as proper times inside the fluid would be much slowed down

compared

^{to the} proper times of external observers.

However,

^{we will not here}

investigate

^the

validity

^{of our}

assumptions,

but will content ourselves with

exploring

^their consequences. It would be

interesting

^{to see} how the methods of this _paper

might

be extended to include the flow of heat and radiation

coupled

^{with the matter fields.}

287

A MAXIMUM ENTROPY PRINCIPLE

We use the

following

notation and conventions. The

expression

^for

the interval

length

will be written as

with Greek indices

taking

^on the values

0, 1, 2,

^{3. Latin indices will be} understood to assume the values

1, 2, 3.

Thus the fluid

4-velocity

satisfies ⁼ 1.

We choose units so that the

velocity

^of

light

^{c = 1 and the Newtonian}

gravitational

^{constant G = 1. Thus we} will write the field

equations

^as

where is the Einstein tensor, which satisfies the conservation

identity

V

03B1S 03B1

⁼

0, ~

^«

denoting

^the

operation

of covariant

differentiation,

^and

is the energy-momentum ^tensor.

The

energy-momentum

^{tensor of the}

perfect

^fluid

^without

^{heat flow is}

then

expressed

in these units as

where e is the total proper energy

density, p

^{is the} pressure, and is the

hydrodynamic

^flow

4-velocity.

. 1. ^- THE DYNAMICAL METHOD

IN GENERAL RELATIVITY

To

bring

into focus some of the ideas of the

introduction,

^{we here}

present

a sketch of the

dynamical

^{method of}

testing

^the

stability

^{of fluid}

spheres

ⁱⁿ

general relativity.

^{We have mentioned that we will}

only

^consider

stability against spherically symmetrical perturbations,

^{and since} any static

configu-

ration of fluid will exhibit such symmetry, ^we way

conveniently

^{take this}

as a

starting point.

It will be useful to

employ spherical polar

coordinates and to define the radius

parameter

^r

by using

the metric form

v and x

being

^{real functions of r and t.}

Einstein’s field

equations Sf

^{= - then}

give ^[8], using

^a

prime

^to

denote

()-/o-r, for l.L

^{= v =}

1,

288 W. J. COCKE

and

for y = v = 0,

where we have used ⁼ 1.

Integration

^of

Eq. (1-3) yields

For y

⁼

1, v

⁼

0, we

^obtain

where the dot indicates The

equation for ,

^{= v = 2 would}

provide

a fourth

equation (There

^{are four}

independent unknowns,

which may be taken to be _e,

ul,

x, ^and v, p

being given

^{as a} function of the other variables.

See section

II). However,

^{it is more} convenient to use

the l-L

^{= 1}

component

of the conservation

equation V

^{= 0. We shall not write it out in}

^full,

but rather refer the reader to the literature

[9].

We now denote

where so, po,

a~,

vo ^are solutions of the static

equations

or

and of the static

part

of the conservation

equation

⁼

0,

which reads

Equation (1-6)

^{is the}

equation

^of

hydrostatic equilibrium,

^{which we will}

derive

again

from the maximum entropy

principle

in section III. It

is,

of _course, the

analogue

^{of the Newtonian}

equation dp/dr

^{= -}

pdp/dr,

p

being

^{the mass}

density and p

^the

gravitational potential.

Following

Chandrasekhar

[7],

^{we assume that}

8e, 8p, 8À, 8v,

^{and u1 are}

289

A ENTROPY PRINCIPLE

small

quantities

^{defined on the} interval 0 ^r

Ro,

^where

Ro

is the boun-

dary

radius of the static

distribution ;

^{and we write to} first order

of smallness

Letting ~(r, t )

^{be the «}

Lagrangian displacement

^{)) of an} element of the fluid from an initial

position,

^{we have to} first order

and

integrating Eq. ( 1-5),

^we

get, with U4

^"’

eVo/2,

From V

a Tla

⁼

0,

^{to first}

order,

^one may obtain the relation

Since this

equation

^is linear in the

dependent

variables and contains

only

the second time

derivative,

^we may

separate

^variables

by assuming

^{a time-}

dependence

of the form cos at, ^a

being

^{a constant,}

getting

where we now consider

8 ~, 8v, 8p, oe, and ~

^{to be} functions of r

only.

In the ^next section ^we show that for adiabatic motions we may

write p

in the

form p

⁼

p(e, x),

^where the variable x indexes the fluid element itself and follows its motion. See also reference

[1].

^{Then we have in}

our case

and we may ^use this relation to substitute for

ap

ⁱⁿ

Eq. (1-7),

^and

use

Eq. (1-2") (1-4")

^and

( 1-5 ")

^to eliminate other

variables, finally writing

Eq. (1-7) ^{with ç}

^as

dependant

^variable

290 W. J. COCKE

where one can check to see that P ~ ^0.

Multiplication by

^the

quantity q(r)

^{= exp allows us to write the form}

where

q(r), v(r)

&#x3E; 0. The

boundary

conditions

on ç

^are

arranged

^{so that}

the pressure vanishes ^to first order ^on the new

boundary

^radius

Ro

+

ç(Ro).

This condition reads

which the reader _may

easily

write himself in terms

of ç.

^{At r = 0 we must} have of

course ç

^{= 0.}

Equations (1-8)

^and

(1-9)

constitute ^a Sturm-Liouville

eigenvalue problem

The

problem

is of the

singular

type

since q

^{and v} may be shown ^to tend to 0 at both ends of the interval 0 ^r

Ro. However,

^one may show

[2]

that in

spite

^{of the}

singular

character of the

problem,

there exists ^an infinite discrete set of

eigenvalues { 6~ },

of which there is a least member

ao-

It is

easily

^shown

[2, 10]

^{that we} may solve

Eq. (1-8)

^{for a2 as}

thus

defining

the functionals

3)[~], JC[~].

^The

boundary

^{terms vanish}

because

q(0)

⁼

q(Ro)

^{= 0.}

According

^{to theorems on the extremum} pro-

perties

^of

eigenvalues [2, 10]

the lowest

eigenvalue 6o

^is

^given by

where 03C6

ranges ^over all functions with

bounded, piece-wise

continuous

p’

which

satisfy

^the

boundary

conditions

for ç

^{and do not vanish}

identically.

The condition

for

stability

^{was first}

applied

^{to this}

problem by

^S. Chandrasekhar

[1].

Note that

since q, v &#x3E; 0,

^{if t ~ 0 for all} 0 ~ ^r

Ro,

^then

G~

&#x3E; 0 would follow

immediately,

and any such

configuration

^{would be}

dynamically

stable.

Having

studied the

application

^{of the}

dynamical

^{method in}

general

^rela-

tivity,

^{we now}

proceed

^to

develope

^our

replacement

^{of the} energy

method,

the maximum entropy

principle.

291

A ENTROPY PRINCIPLE

2. ^- THE TOTAL ENTROPY

In the introduction ^we stated the need for

expressing

^{the total}

entropy

^S of the fluid

sphere

^{in terms} of variables

already entering

into the field _equa- tions. To

accomplish

^this

end,

^{we now construct a} relation between the proper entropy

density s

per unit volume and the pressure p and _energy

density

^e.

First let V be the volume of a very small element of the fluid. Since we

consider all motions to be

adiabatic,

^{the total}

entropy

^{S = sV} of the ele- ment will be constant over such

motions,

^{and we} may write the

thermodyna-

mic

identity

for the element as dE +

pdV

^{= TdS}

^{= 0;}

^{or, since E =}

sV,

Now since a

change

in volume is the

only

^{means of}

changing

the thermo-

dynamic

^state of the fluid

element,

^{heat flow}

having

^been

outlawed,

^it

follows that _any

change

^{in state will be}

accompanied by

^{a non-zero}

change

in _e,

except

for elements in

which p

^{and e are both zero}

simultaneously.

Thus, except

^{for such «}

empty » elements,

^we may say all the

thermodynamic

variables may be written ^as functions of the _energy

density

^e,

including

of course the pressure.

However,

^the

equation

^{of state} p ⁼

f(e)

will then in

general

^{be a} different function of e for different fluid elements.

Let us now consider the use of a

co-moving

coordinate

system

^{in which} the

spatial components

^uj

(~’ = 1, 2, 3)

of the fluid

4-velocity

^vanish

throughout

^the

system

for all time. Such coordinates ^are

presumably always possible

^since

they

may be defined

by simply attaching

^the

spatial

reference

system

^to the identifiable

points

of the fluid.

By

virtue of the above arguments ^we may ^now write the pressure ^{as a} function of e and of the

spatial

coordinates xj of the

co-moving

^frame:

p ⁼

p(e,

^{the xj}

serving

^to index the different

points

^{of the}

fluid ;

^i. e., the different fluid elements. It must be undertood that the function

p(e, xi)

is to be derived from

purely thermodynamic

considerations after

specifying

the initial _pressure and _energy

density

distributions at a time x° =

But since the total entropy of the fluid element S ⁼ sV is

conserved,

^we

may form d S ⁼

d(sV)

^{= sdV + Vds =}

0,

^{or -}

dV/V

⁼

ds/s. Hence,

from

above,

292 W. J. COCKE

Integration

^{with xi = constant}

yields

or

where _so and _so are functions of xj such that

s(eo)

^{= so.}

We have thus

expressed

^the

entropy density s

^{as a} definite function of the _energy

density e

^{and the}

spatial

co ordinates xj of the

co-moving

^{frame jc~.}

Let it be

emphasized again

^{that this}

expression

^{is valid}

only

for adiabatic

changes

^{of state.}

Note that of it is

possible

^{to define a conserved}

particle

^{number for the}

system ^{such that n is the} proper

particle

^number

density,

^{the total}

particle

number N ⁼ nV of ^a small fluid element would also be

conserved;

^and

likewise, - d V/V

⁼

dn/n.

^{Hence we would}

similarly

^have

°

and thus n and s would be

proportional

^{to the same function of s for fixed xj.}

As an

example

^{let us find an}

expression

^{for the}

entropy density

^{of a} per- fect non-relativistic Boltzmann _gas with constant

specific

^{heat. We} may write the

equation

^{of state for an element of such a} gas ^as

pV

⁼

NkT,

or p ⁼ nkT. For adiabatic

compression

^or

expansion

^{we further}

have,

with _y as the constant ratio of

specific

heats per

particle

y ⁼

cp/cv ^[11], pVY

^{= constant} or p ^{= constant} nY, ^{or n = where} no and _pa

are functions of xj. Then the _energy

density

^is

given by

where mo is the rest mass per

particle

^and Eo is an

arbitrary

^{constant. There-}

fore

which definies _p ⁼

p(e, xj)

^{as an}

implicit

function. Then

where we have used cp ^{- Cy =} k. It then follows that

We have thus verified the

proportionality

^{of sand n.}

293

A MAXIMUM ENTROPY PRINCIPLE

The total

entropy

of the fluid

element

_may be written

[11]

where ~

^{is the chemical constant of the} gas. Hence

This verifies the

expression

for s derived above from

Eq. (2-1).

We must now show that our

expression

^{for s satisfies the} proper ^conser- vation

equation

^{= 0. Let us} write the scalar in terms of

our

co-moving

frame in which

only

^{u° is}

non-vanishing :

where indicates Now we have from

Eq. (2-1)

^that

so that

and hence

But this is ^a tensor

equation,

and hence is true not

only

^{in the}

co-moving

frame but _{in any}

system

of coordinates whatever. Now the field

equations imply

^that

Let ^us further construct the scalar

product

um v ⁼

0, using

^{the rela-}

tion ⁼ 1 and its

consequent

^{= 0 :}

Comparison

^with

Eq. (2-3)

then shows that in

fact,

if the field

equations

are ⁼ 0.

294 W. J. COCKE

We may ^use this relation to show that the total _proper

entropy

^{of the} fluid

system

^{is a conserved}

quantity.

^{Let t be a} time-like coordinate and x, y, ^z be

space-like

coordinates

(not necessarily

^{Galilean at}

infinity).

^Let

2p

^{be the}

space-like

surface of the mass of

fluid,

^{and let}

~a

^and

~b

^{be the} intersections of the time-like surfaces t = a and t

= b, respectively,

^with

the

space-time

volume of the fluid. Then if

F

^{be the}

part

^of

03A3F

^which lies between

~a

^and

~b,

the surfaces

~Q, ~b,

^and

2p together

^{form a closed}

3-space E

ⁱⁿ

space-time.

Now let V denote the

space-time

volume of fluid enclosed

by E.

^Then

for an

arbitrary

^{vector field}

A~,

^we may write Green’s theorem for this volume and surface as

[12]

where n~ is the outward unit normal ^to

~,

^{and and are}

generalized

3- and 4-volume elements.

If

^{= 0 is the}

equation

^of

^I:,

^{then we have}

with the

sign

^{chosen so} that n~ is directed outward from E. Inside V

we can let ⁼

il’ -

^g

dt dx dy dz

^{and on}

Ea

^and

~b

^{we can set}

=

Ý -

^{g~3~ dx}

^{dy dz,}

^where is the determinant of the subtensor gij

(i~ .I

⁼

1, 2, 3).

Then if we choose A~ ⁼ we will have ^{= =} 0 inside

V,

and

consequently, by

^Green’s

theorem,

Now the

equation

^for

~a is

^{= t - a =}

0,

^{so that on}

E.,

^{for a}

b,

Similarly,

^on

^1~

Thus it follows

that,

^since

295

A ENTROPY PRINCIPLE

Further,

^{we must also}

satisfy

^the Lichnerowicz

junction

conditions

[13],

from which one may

easily

deduce that ^on the surface

2p,

ua.na. ⁼ 0. This

can also be shown

by expressing

^u~ and ntL in

co-moving coordinates,

ⁱⁿ

which the

equation

^for

Ep

will have the

form f(xj)

^{= 0.}

Thus we obtain

and hence

for all t = a, b.

Or,

where

2~

^{is the}

3-space t

^{= constant. The}

integral

^S may be said to repre- sent the total

entropy

^{in the} same sense that conservation of

charge

^{in elec-}

trodynamics

in the form ⁼

0,

^where p is the

charge density,

^{leads to}

an

exactly

^similar

expression

for the total conserved

charge.

We ^now

proceed

^{to formulate our} variational

principle

^{S = maximum}

with

respect

^to variations which conserve the total _energy.

3. ^- THE FIRST VARIATION

We now

require

^{that for}

hydrostatic equilibrium

the total

entropy

^S be an extremum

(maximum)

for all infinitesimal adiabatic variations of the energy

density e

and pressure p which ^conserve the total _energy of the _sys- tem. As

stated,

^we will restrict ourselves to variations which _preserve the

spherical symmetry

^{of the}

fluid,

^{and thus we} will be able to use the expres- sions for the metric form and for the field

equations presented

in section I.

It should be stated at the ^outset that in

performing

the variations we

will not consider the variations as

independent

from ~e and

8p,

^but

will rather let them be

given

^{via the field}

equations (1-2") ( 1-4")

^and

(1-5").

We shall not use the

equations T2

^{= -}

803C0S22

^{and =}

^0,

^since

they

themselves

imply

^the

equation

^of

hydrostatic equilibrium. Further,

^the

variation

öp

will be linked to ~s

by

^means of the adiabatic

equation

^{of state}

p ⁼

p( e:, representing,

^{as it} were, the

point

^of

origin

of the fluid element. Thus ^we will have

only

^one

independent

variational function.

296 W. J. COCKE

Let us also note that even

though

the variations are

adiabatic, they

^will

not conserve the total

entropy identically,

^{since we will}

always

consider the

spatial velocity

of the material to be zero.

In order to carry ^{out our} variations ^we will need an

expression

^{for the}

total _energy. It is

easily

shown that for the metric from

(1-1),

^{a conserved}

energy may be written in the forms

[14]

Thus for the static

problem

^{we take ui = 0 and write}

Using Eq. (2-4),

^one may then _express the total

entropy,

^after

performing elementary integrations,

^as

But with u1 ⁼

dr/ds

⁼

0,

^{we have from}

Eq. (1-1)

^that

dt/ds

⁼

Hence

However,

ⁱⁿ

taking

the variations of M and

S,

^{we must} allow for the fact that the

boundary

^{radius R will also} vary. Thus it behooves ^us to trans- form the

independent

variable r so that the condition M ⁼ constant auto-

matically

fixes the end

points

^of the interval of

integration.

^{We do this}

by transforming

^to the variable

which

represents

^the

quantity

^of energy inside the radius r.

We denote

dr/dm by

^r

(we

^trust there will be no confusion with a time

.

derivative in this

context)

^{and find}

dm/dr

^{= or = r =}

and thus

Also we have

297

A ENTROPY PRINCIPLE

We _may hence _express the

integrand

^{of S as a} function of the

independent

variable ^m and the

dependent

^{variable r and its} derivative r. The

princi- ple

^{~S = 0} will then determine r ⁼

r(m)

from the Euler

equation,

^{and the}

condition M ⁼ constant will be

equivalent

^to

fixing

^the upper end

point

of the interval of

integration

^{0 m M.}

However,

^{careful note must} be made of the fact that the

dependence

^of

the

entropy density s

⁼

xi)

^{on the}

spatial

variables xi is such that xi

always

follows the motion of the fluid element. Thus in

reality,

^{we have}

for the

spherically symmetric problem s

⁼

s(e,

^{r -}

Q, where ~

is the dis-

placement

of the element of fluid which finds itself at r after

having

^been

moved from the

point

^{r -}

ç. ç

^{is then}

given

^{in terms of the} other varia- bles

through

the relation

( 1-5 ").

^{Let us} denote ~ == r ^-

ç,

^{so that}

s ⁼

x).

Our next

step

^{would be to} write the variation of S in terms of the varia- tion or ⁼ of the

dependent

^{variable r.}

^However,

^{we must first be}

able to

express ~

^{in terms of Sr. Here} is the infinitesimal

change

^of

the radius r ⁼

r(m)

^{of a}

sphere

of fixed energy content m, and not the

Lagrangian displacement,

^{which we have}

designated

^as

^~.

Let us denote the

change

ⁱⁿ energy ^content

m(r)

^{of a}

sphere

^{of radius r}

by

⁼

403C0r0

03B4~r2dr. Then

writing mo(r)

^{as the}

equilibrium

distribution

0

function around which the variation is

taken,

^{we have}

by

^definition

and hence

to first order.

Thus, using Eq. ( 1-4")

^and

( 1-5 "),

^{we obtain}

We now write

298 W. J. COCKE

The first variation of S will thus be

expressed

^as

According

^to the fundamental lemma of the calculus of variations the condition 3S ⁼ 0

implies

^{the Euler}

equation

and the condition ⁼ 0 since in

general ~r(M) 5~ 0, although

^{of course}

8r(0)

^{= 0.}

We first

investigate

^the

boundary

^{term : We have}

and

using Eq. (3-3),

^which

gives

^{= -}

(403C0r2r2)-1

^{== -}

~/r

^{one obtains}

easily Lr - e03BB/2r2sp(p

⁺

e)-I.

Thus if the _energy

density

^{does not vanish at the}

surface,

^{we will have}

= 0

through

the condition that the _pressure on the surface vanish.

But suppose e ⁼ 0 also. Then we may write

Lr -

^and

demand

simply

^that remain finite as e - 0. This is

certainly

^a very reasonable

stipulation,

^{and one can}

show,

^for

example,

^{that for}

equations

of state of the form e = _«p ~--

(see Eq. (2-2)), stays

^{finite as} e2014~0 for

positive

^«,

p,

y if and

only

^if y &#x3E; 1.

However,

^one may also show that the

product

goes ^{to zero as e} --~ 0 for all _y &#x3E;

0,

^{so that}

Lr]M

^{= 0}

^still

^{tells us} very little about the

equation

^{of state.}

We now

proceed

^to examine the Euler

equation (3-6).

^{Let us write}

d(Lr )/drn

⁼

(dr/dm)d(Lr )/dr

^{= and} express

Eq. (3-6)

^as

a total differential

equation

^{with r as}

independent

^{variable as}

before, again using

^a

prime

^{to denote}

d/dr.

^{We have}

where

Eq. (3-3)

^and

(3-4) give

299

A ENTROPY PRINCIPLE

and

Further

But

so that the terms in will cancel. Then after

using Eq. (1-3’)

^and

collecting

^{terms, we may write}

Eq. (3-6)

^as

which

^with the aid of

Eq. (1-2’)

may be written

p’(p

^{+ +}

v’/2

⁼

0,

which ^we

recognize

^as

Eq. (1-6),

^the

equation

^of

hydrostatic equilibrium.

Thus ^we have obtained

Eq. (1-6) by

^means of the maximum

entropy principle

and with the

help

^{of the}

equations S~

^{= -}

8?rT~ S~

^{= -}

and

S~

^{= - We now}

proceed

^to examine the second variation ~2S and

require

that it be

negative.

4. ^- THE SECOND VARIATION AND

EQUIVALENCE

WITH THE DYNAMICAL METHOD

Now that we have found ^a _necessary condition

(the

^Euler

equation)

^{for S}

to be ^an extremum, the

principle

^{S = maximum}

requires

^{us to}

investigate

82S 0 for infinitesimal variations about the extremum. We will show that this

requirement

^is

equivalent

^to be criterion

developed by

^Chandra-

sekhar

by

^{means of the}

dynamical

^method. It will be shown in fact that if

~o

be the least

eigenvalue satisfying Eq. (1-8),

^then

ao

&#x3E; 0 if and

only

if a2s 0 for all non-zero infinitesimal variations

conserving

^total energy and

satisfying

^{the usual}

continuity

^and

boundary

conditions.

We will use

again

^the

independent

^{variable m as}

developped

in the last section to accommodate variations in the

boundary

^{radius R and to}

comply automatically

^{with M = constant}

by fixing

^the upper end

point

of the inter- val of

integration.

ANN. INST. POINCARÉ. A-II-4 20

300 W. J. COCKE

The second variation 82S is

given by [6, 16]

Substitution of ~r

for ~

^via

Eq. (3-5)

then allows us to write 82S in the form

thus

defining

^the

quadratic

^form

Q(8r, 03B4r).

A rather

simple

^{test to} be satisfied for S to be a maximum is the so-called

Legendre-Weierstrass

^{test, which is a necessary} condition for 828 0 with

respect

^{to both}

strong

^{and weak} variations. It is a local test and is writ- ten

[16, 17]

where rand

r

are actual extremal

values, p

^{is any}

physically

realizable value of r, and 6 is _any number between zero and one. The

equality sign

^can

hold

only

for isolated

points.

But

where the

dependence

^on

r

is

through

^{£ =}

r),

and hence for

positive

^£

the test is satisfied if &#x3E;

0,

^which

certainly

^{holds for} any real one-

phase

^fluid. This also shows that S

certainly

^{cannot be a minimum.}

We now

proceed

^{to formulate a more}

general

^test

(related

^to the so-called Jacobi test

[18])

which includes the

Legendre-

Weierstrass test in its formula- tion and which we will show is

equivalent

^to Chandrasekhar’s criterion.

First we demonstrate that a necessary and sufficient condition for ö2S 0 is that all the

eigenvalues ~

of the associated linear

problem (abbreviating

etc.)

must be

positive.

^The

boundary

conditions to be

applied

ⁱⁿ

finding

the are

given

^as

the latter of which _may be shown to be the same as

Eq. (1-9),

^{which states}

that the pressure ^must vanish to first order ^on the

varying boundary

^{of the}

sphere.

301

A MAXIMUM ENTROPY PRINCIPLE

Now let us

multiply Eq. (4-1) by

^{or and}

integrate, getting

Again exploiting

the extremal

properties

^of

eigenvalues,

^one may

easily

show that for the smallest

eigenvalue

^ll-o,

for all

admissible 03C6

^not

identically

^zero

satisfying

^{the same}

boundary

^and

continuity

conditions as ~r. Thus if _yo &#x3E;

0,

^{then 82S}

0,

^and

conversely.

We are now in ^a

position

^{to compare our} condition 82S 0 with Chan- drasekhar’s

criterion,

^{which as we} have said results from

linearizing

^the

time-dependent equations

^{of motion.}

Substituting ~

for Sr in

Eq. (4-1)

with the use of

Eq. (3-5),

^{one may, after}

changing

^the

independent

^variable

back to r and

performing

very tedious

manipulations,

^{obtain an}

equation

of the form

where q, u &#x3E; 0.

Comparison

^with

Eq. (1-8),

^the

eigenvalue equation

for a2 in Chandrasekhar’s

criterion,

^{would show that}

q(r)

^and

t(r)

^{are the}

same

functions

^{in both}

equations,

but that in

general u(r) ~ v(r).

^The

same

boundary

conditions of course hold for both

equations.

We will now prove that all the

eigenvalues

^{l1. of}

^Eq. (4-2)

^are

positive

if and

only

if all the a2’s in

Eq. (1-8)

^are

positive.

^As

before,

^{we write}

for the lowest

eigenvalues

f1.o and

~o, ^using q(0)

⁼

q(R)

⁼

0,

and

the functional

being

^{of course the same in both}

equations.

^{Now one}

can show

[2]

^{that there exists an} admissible cpo such that

302 w. J. cocKE

Now _suppose 0.

0,

^{and hence}

Therefore 0

implies ao ^0,

^{and one may}

^similarly

^{show that}

0

implies

^{0. Hence all the}

[.L’S

^are

positive

^{if and}

only

^{if all}

the a2’s ^are

positive,

^{and hence 82S 0 if and}

only

^if

cr:

&#x3E; 0.

This shows the

equivalence

^{of the two}

stability

criteria. We now pass

on to consider

applications

^{of our formalism to} certain features of actual models.

5. ^- APPLICATION TO FEATURES

OF ACTUAL MODELS

Actual

equilibrium

models for a

variety

^of

equations

^{of state have been}

fairly thoroughly investigated,

^{and the}

stability

^{of some of these models has}

been tested with Chandrasekhar’s criterion

[19].

^{Since the} results which

we will discuss

[19, 20]

^were obtained for

equations

^{of state of the form}

p ⁼

peE)

^{with no}

dependence

^{on a}

co-moving

^variable x, ^we will _{use p} ⁼

p(e) throughout

this section as a

simplifying assumption.

One

aspect

of the results obtained takes the form of ^curves of total energy

(mass)

of the models as a function of ^a

parameter

^{such as the central} pres-

sure p~. We write M ⁼

MoCPc).

^One may show that

only

^{one such}

function

Mo(pc)

^is obtainable for a

given equation

^{of strate.}

A

stability analysis

^{for a}

given pee)

then shows which values of _pc ^are associated with stable

configurations

and which with unstable ones. One feature that

usually

appears is that for ^a

given equation

^{of state there is}

a

point ~

such that for 0 ~ _pc

p°

^the

configurations

^are

stable,

^while

for _pc &#x3E;

p~

^the

configurations

^are unstable. We call the

points p~

^{« tran-}

sition

points )). Further,

the transition

points po

^{are known to be « mass}

peaks

^{», or} maxima of the functions

Mo(pc). However,

^it appears that not all mass

peaks

^{are such} transition

points.

^{We will} prove ^a

plausible

^suffi-

cient condition for a mass

peak

^{to be a}

point

of either neutral or unstable

equilibrium.

^{This will}

apply

^{as well to minima on the mass curves.}

Oppen-

heimer and Volkoff

[20]

^have

presented

arguments which lead one to the conclusion that all maxima and minima must be such

stability-instability

transition

points.

Calculations

by

Misner and

Zapolsky [19]

^{have shown}

that this is not the case, and we will

explain why Oppenheimer

^{and Vol-}

koff’s

argument

^fails.